I have 10 incidents associated to a problem and it gets very monotonous to update all the incidents with workarounds/solutions while closing the problem. Is there a better way to handle this ?

You can automate in such a way that when a solution is added to the problem it automatically gets copied to every incident which is associated to the problem.
This can be done under Admin -> Problem Closure rules -> Copy problem solution and workaround to all associated incidents.

When I submit a change, request approval, get approval, manually change the status of the change to "Approved" the change still appears on my "unapproved changes" widget on my home page in the helpdesk tool. Am I missing something here ?

Only the changes approved by the Change Manager are listed under Approved Changes. According to ITIL, the Change Manager decides whether to approve or reject a change.

The status of all the changes I create are Approved automatically, what is the reason ?

Standard changes are pre-approved changes. Make sure you do not select the Change type to be Standard unless the change does not need approval from the Change manager.
In 7.6 we have an option to customize standard Change to whether it should be approved or not.

Can a change be implemented if the CAB rejects the change and the Change Manager approves it ?

Yes, the change can still be implemented as long as the change manager approves it.

Any change I create does not get reflected in my change calender, even though I mention the start time and end time.

The change schedule will reflect in change calendar only after it is approved by the change manager.